A homomorphism of Lie algebras is a linear map such that for all we have
Given a commutative unital ring , and a (strict for simplicity) symmetric monoidal -linear category with the symmetry , a Lie algebra in is an object in together with a morphism such that the Jacobi identity
hold. If is the ring of integers, then we say (internal) Lie ring, and if is a field and then we say a Lie -algebra. Other interesting cases are super-Lie algebras, which are the Lie algebras in the symmetric monoidal category of supervector spaces and the Lie algebras in the Loday-Pirashvili tensor category of linear maps.
Lie algebras are equivalently groups in “infinitesimal geometry”.
More generally in geometric homotopy theory, Lie algebras, being 0-truncated L-∞ algebras are equivalently “infinitesimal ∞-group geometric ∞-stacks” (e.g. here), also called formal moduli problems (see there for more).
Notions of Lie algebras with extra stuff, structure, property includes
Examples of sequences of local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||integers|
|Lie theory||Lie algebra||formal group||local Lie group||Lie group|
|symplectic geometry||Poisson manifold||formal deformation quantization||local strict deformation quantization||strict deformation quantization|
Discussion with a view towards Chern-Weil theory is in chapter IV in vol III of
Discussion in synthetic differential geometry is in